Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known F_q² -maximal curve X₃ having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough X₃ has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than its set of F_q² -rational points. This makes the curve X₃ the first explicitly known maximal curve having non-rational Weierstrass points. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(X₃) is exactly the automorphism group inherited from the Hermitian curve, apart from small values of q.